University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Practice Exercises - Page 204: 126



Work Step by Step

$$A=\lim_{\theta\to0}\frac{\sin(\sin\theta)}{\theta}$$ $$A=\lim_{\theta\to0}\frac{\sin(\sin\theta)}{\sin\theta}\times\lim_{\theta\to0}\frac{\sin\theta}{\theta}$$ We have $$\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$$ For $\lim_{\theta\to0}\frac{\sin(\sin\theta)}{\sin\theta}$, take $\sin\theta=x$, then as $\theta\to0$, we have $x=\sin\theta\to\sin0=0$ So $$\lim_{\theta\to0}\frac{\sin(\sin\theta)}{\sin\theta}=\lim_{x\to0}\frac{\sin x}{x}=1$$ Therefore, $$A=1\times1=1$$
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